We consider the Klein-Gordon equation on a star-shaped network
composed of $n$ half-axes connected at their origins. We add a
potential which is constant but different on each branch. The
corresponding spatial operator is self-adjoint and we state explicit
expressions for its resolvent and its resolution of the identity in
terms of generalized eigenfunctions. This leads to a generalized
Fourier type inversion formula in terms of an expansion in
generalized eigenfunctions. This paper is a survey of a longer
article, nevertheless the proof of the central formula is indicated.